Video Math Tutor: Basic Math: Lesson 3 - Operations on Numbers
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BA S I CMA T HASelf-Tutorial
by
LuisAnthonyAstProfessionalMathematics Tutor
LESSON3:
OPERATIONSONNUMBERS
Copyright2005All right s reserved. No par t of th is publicat ion ma y be reproduced or tra nsm itted in an y form
or by an y mea ns, electronic or mecha nical, including ph otocopy, recordin g, or an y inform at ion
storage or retr ieval system, without per mission in writing of the au thor.
E-mail may be sent [email protected]
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=Disclaimer: T h i s Se lf-T u t o r i a l is m e a n t f or s t u d e n t s ( 7t h g r a d et h r o u g h c o lle g e ) w h o n e e d a b a s i c r e v ie w o f a r i t h m e t i c on
i n t e g e r s (p o s it i ve a n d n e g a t i ve w h o le n u m b e r s ) a n d O r d e r o f
Op e r a t i o n s . I t i s NOTm e a n t fo r y o u n g c h i ld r e n w h o a r e
le a r n i n g a r i t h m e t i c fo r t h e fi r s t t i m e .
|ABSOLUTE VALUE|FTh e Absolute V a l u e of a nu mber is t he distan ce thatnu mber is fromzero on t he r eal nu mber line.
Mat h S y m b o l:
L The a bsolute value of a nu mber is NEVER n egative. Itseith er positive or zero.
Form al definit ion of Absolut e Value ofx:
This definition m eans th at th e absolute value of a real n umber x is ju st
itself if th e nu mber is zero or positive, and you cha nge t he s ign if th e
nu mber is negat ive. The x does n ot mea n nega tive x, but r at her t he
opposit e ofx. This is discussed in more deta il a litt le lat er on.
EXAMP LE 1: Find .
SOLUTION: This is h ow you can see wha t t he a bsolut e value ofth e number 3 is:
1 unit + 1 unit + 1 un it = 3 units a way from 0
| |||
d||d0 3
1444442444443
The absolute value of 3, denoted by is equal to th e nu mber of un its it is
awa y from zero, since it is th ree un its awa y, th en .
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F The a bsolut e value of 3 is also 3, since 3 is t hr eeun its a way from zero:
1 unit + 1 unit + 1 un it = 3 units a way from 0
| |||
d||d3 0 1444442444443
Together:
3 un its awa y from 0 3 un its awa y from 0
| | |
d||d||d
3 0 3 14444424444431444442444443
EXAMP LE 2: What is ?
SOLUTION: becau se 4 isfourun its a way from zero.
d|||d
0 4 1444442444443
four un its away
EXAMP LE 3: What is ?
SOLUTION: since 0 is well ZERO un its awa y from itself,
it is th e ONLY absolut e value t ha t is equal t o zero. All oth ers ar e positive.
d
0
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OPPOSITENUMBERS
FOpposi te Number s ar e th e same distan ce from zero on t he r ealnu mber line, but th ey are on opposite sides on t he n um ber line.
Mat h S y m b o l: or ( )
EXAMP LE 4:
3 is th e opposite of 3 since th ey ar e both th ree
un its a way from zero, but ar e in opposite directions.
d|||||d
3 0 3 % Opposite Num bers &
L Opposite nu mbers h ave th e sam e absolute value, so
EXAMP LE 5:
Y Wha t is t he opposite of 2? A n s w e r : 2
Y Wha t is t he opposite of 5? A n s w e r : 5
Y Wha t is t he opposite of? A n s w e r :
Y Wha t is t he opposite of 0? A n s w e r : 0
A nega tive sign in front of a n um ber, var iable, or gr oup ing symbol is used
to symbolize the opposite. So
x is t he opposite ofx
(8) is the opposit e of 8, which is 8
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( th e nega tion key is used to find th e opposite ofa n um ber of an expression; so it is used to represent both a negat ive
nu mber a nd t he n egation (the opposite) of a n um ber.
Wh a t t o d o: On t h e Ca lcu la t or S creen :
Fin d t he opposite of 2
M
Or M
=Be sur e you use th e negation key an d NOT th e subtr action k ey
when you wan t t o find th e opposite of some expression.
F Opposites a re a lso ca lled Add i t i ve Inverses(more on t his in Lesson #4).
GGGGGGGGGGGGGGGGGG
ARITHMETICOF INTEGERS
A+D+D+I+T+I+O+NFAddi t ion is th e opera tion of combin ing nu mber s to provide anequivalent single value. The n um bers being added ar e called the
summ an ds and th e result is called the sum.
S u m m a n d + S u m m a n d = S u m
Mat h S y m b o l: +
Addition of nu mbers can be visua lized on t he n um ber line. Sta rt at th e
origin , th en move to th e right, if the n um ber is positive, an d left if
th e num ber is negat ive.
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EXAMP LE 6: Add 2 + 3 on a nu mber line.
Step 1: Go 2 un its t o th e right , since 2 is positive:
2 units
||||||||
0
Step 2: Go 3 un its t o th e right , since 3 is positive:
3 units
||||||||
0 2
Since we stopped at 5, tha t is our su m, so 2 + 3 = 5.
Lets t hr ow in a n egative num ber n ow.
EXAMP LE 7: Add 3 + 4 on a nu mber line.
Step 1: Go 3 un its t o th e left, since 3 is n egat ive:
3 units
||||||||
0
Step 2: Fr om 3, go 4 un its t o th e right , since 4 is positive:
4 units
||||||||
3 0
Ther efore, 3 + 4 = 1
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The pr evious example ma y also be done on a single num ber line:
||||||||
3 0 1
3 + 4 = 1
It is cumber some t o always u se th e nu mber line for a ddition, so we
genera lly just add n um bers m ent ally, use our fingers an d toes, or s imply
use a calculat or. We use th e following ru les to help us with addit ion:
RULES OF ADDITION
R u l e 1 : positive + positive = positive
Add t he n um bers, th e sign of the su m is a lso positive.
R u l e 2 : negat ive + negat ive = negat ive
Add th e nu mber s, the sign of th e sum is also negat ive.
Rules 1and2 mea n: if you ha ve like signs, add up th e nu mber s,
an d place th e sign you see for th e final su m.
R u l e 3 : positive + negative or negative + positive
This r ule is a little more complicat ed
What you do is tem pora rily ignore t he signs you see, subtr act t he sma ller
nu mber from t he larger, th en include t he sign of th e nu mber whose
absolut e value was bigger (whichever nu mber is bigger with out looking a t
th e plus/minu s signs, is the s ign you will use for t he fina l resu lt.) Dont
worry, we will do a few examples of th is to mak e it eas ier to lear n.
R u l e 4 : any nu mber + its opposit e = 0
This is called th e Add i t i ve Inverse proper ty of addition. The
opposite is th e additive inverse of a nu mber .
R u l e 5 : 0 + a n um ber or a nu mber + 0 = th e nu mber
This is called th e Add i t i ve Ident i ty proper ty of addition. Zero
is the a dditive ident ity.
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EXAMP LE 8: Her e a re some m ini-examples. Try doing th em by
ha nd first , then use your calculat or.
One of th e keys to su ccessfully using your ca lcula tor is t oPRACTICE, PRACTICE , an d PRACTICE . Yes, th ese are simple problems,but m y recomm enda tion is to always tr y to do problems by han d firs t (if
possible), then use a calculat or . Why? Well, if you only use your
ca lcula tor for th e tough problems, you m ay not ha ve th e experience to do
th e problem. Stu dent s ma y blank when t rying to find th e right
keystrokes t o do a problem. The m ore you pra ctice, the bett er you get.
(Wh a t t o d o: On t h e Ca lcu la t or S creen :
Exam ple ofR u l e 1 :
Find 8 + 5
M
Exam ple ofR u l e 2 :
Find t he su m of 8 and 10M
Exam ple ofR u l e 3:
Add 12 an d 6
M
Another example ofR u l e 3 :
What is five plus n egative four ?M
Exam ple ofR u l e 4 :
Find t he su m of 9 an d its opposite.M
Exam ple ofR u l e 5 :
What is 0 + 7?M
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L When using a calculat or , you dont n eed to enclose the
negat ive num ber within pa ren th eses, but you m ay do so,
for clar ity, when wr iting it out on pa per . This way, you
dont confuse t he n egative sign with th e su btr action
sign.
Some of our previous problems could be writ ten as follows:
5 + 4 can be writ ten as : 5 + (4)
8 + 10 is th e sa me a s: (8) + (10)
And
9 + 9 is equ al t o 9 + (9)
GGGGGGGGGGGGGGGGGG
SUBTRACTIONFSub t r ac t i on is th e inverse operat ion of addition. Withaddition, you add numbers together, while with subtraction, you take
away numbers.
Mat h S y m b o l:
The nu mber t ha t is ta ken a way from th e origina l num ber is called the
S u b t r a h e n d , the original nu mber is called th e Minuend , an d the result
of a su btr action is called th e Dif ference .
Minuend Subtrahend = Difference
RULE FOR SUBTRACTION
To subtr act one nu mber from another, substitut e the subtr ah end
by its opposite, then ADD th e numbers togeth er.
By doing this, all subt ra ctions become a dditions.
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EXAMP LE 9: Per form t he following operat ions :
Y What is 10 3?
S o l u t i o n : 10 3 = 10 + (3) = 7
Y Subt ra ct 5 from 8.
S o l u t i o n : 8 (5) = 8 + (5) = 8 + 5 = 13
( To subt ra ct u sing a calculat or, use th esubt ra ction key: to perform th e opera tion. Do not mix up this key with
th e nega tion key: . If you do, th is is the er ror messa ge you will see:
If you see th is, press to go to th e err or. The screen cha nges back, butth e cur sor will blink over th e err or. J ust corr ect it, an d press . This
should solve th e problem.
Lets do a s imple subtr action:
Wh a t t o d o: On t h e Ca lcu la t or S creen :
Find : 11 (6)M
N o t e : Look carefully at th e minu s a nd n egative signs on t he calculat or
display. They ar e slightly differen t. The n egative sign is sma ller a nd
slightly higher on t he calculat or display th an th e minus s ign. In t hese
Notes, I follow stan dar d ma th nota tion a nd u sua lly make th em th e same
(but I m ight cha nge th em a litt le, like using b o ld t y p e for emphasis).
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HOT TIP!
When writing negat ive an d minus signs by ha nd, mak e
th em longer th an you h ave in t he pa st. This way, you a re
clear th at a n um ber is negat ive or a subtr action is being
perform ed. I ha ve foun d over t he years th at stu dents t ha t
write very short signs can confuse t hem with decima l
points, t he = sign, or n ot even kn ow th ey ar e t her e.
E x a m p l e : Write inst ead of , which looks like:
GGGGGGGGGGGGGGGGGG
|DISTANCEBETWEEN|
|TWONUMBERS|FWha t if we would like to figure out how far one n um ber is away froman oth er? How can we do th is? For example, I want to find t he dista nce
between 3 an d 4. This is easy to find u sing a n um ber line:
||||||||
3 0 4
J ust coun t t he nu mber of un its between 3 and 4:
||||||||
3 0 4
7 Units
You can coun t st ar ting a t 3 or from 4. It doesnt m at ter .
At t he beginn ing of th is lesson, we lear ned t ha t a bsolute value is u sed tofind t he dist an ce a nu mber is awa y from zero. Now we can combine t his
with our kn owledge of subtr actions, t o get a special form ula th at will help
us find t his dista nce.
The dista nce between 3 an d 4 is =
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In general:
The distance between n um bers a and b on a
nu mber line is:
F
=Do not mix up th is dista nce form ula for t he one t ha t is used to
measu re t he distan ce between t wo points in a plan e:
This form ula is discussed in an oth er lesson.
GGGGGGGGGGGGGGGGGG
MUL TI PLI CA TIONFMul t ip l icat ion is a sh ort -ha nd n ota tion for r epeat ed addition orsubtraction.
Mat h S y m b o l:
For examp le, 2 +2 +2 +2 +2 = 10 can be re-writt en a s a mu ltiplicat ion:5 2 = 10
The symbol is fine to use wh en you ar e just mu ltiplying n um bers
togeth er, but when algebra is involved, it can be confused with th e
varia ble x.
There ar e severa l ways to represen t m ultiplicat ion. Let a and b
represent arbitrar y numbers. a t imes b can be repr esent ed by an y ofthe following:
N ot a t ion : Com m en ts:
ab Not used very much in a lgebra [See above].
abA raised dot. Not usua lly used with n um bers , since it
ma y be confused with a decima l point . OK to use with
variables.
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(a)(b)
[a][b]
{a}{b}
When th ere is NO symbol between nu mber s an d variables,
or jus t va r iables, th en t his is a wa y of showing implied
mu ltiplicat ion. H ere, expressions ar e placed
within a set of grouping symbols, but th ere is nothin g
between groups of symbols. This is a preferred notat ion.
ab
Also implied mu ltiplicat ion. The var iables a re just next to
each other . This is th e most typical wa y of showing th isopera tion with var iables. Not u sed with nu mber s.
(a)b Im plied mult iplicat ion. Not used very often .
a(b)
Im plied mu ltiplicat ion. Th is is frown u pon for
mu ltiplica tion of var iables, since it can be confused with
anoth er m ath nota tion ca lled function nota tion. It s OK
to use with n um bers, or n um bers (in place ofa) with
var iables (in p lace ofb).
The n um bers/var iables being multiplied togeth er a re called Factors , an dth e resu lt of a mu ltiplicat ion is called th e P r oduc t .
(Factor) (Factor) = Product
RULES OF
MULTIPLICATION
R u l e 1 : positive posit ive = posit ivenegative negative = positive
The pr odu ct of nu mber s with like signs is positive.
R u l e 2 : positive nega t ive = nega t ive or
negative positive = negative
The pr oduct of nu mbers with differen t signs is negat ive.
R u l e 3 : 0 an y nu mber = 0 or
an y nu mber 0 = 0
This is called t he Zero-Factor proper ty of mult iplica tion.
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These Rules can be abbreviat ed as:
(+)(+) = + ()() = +
(+)() = ()(+) =
(0)(#) = 0 (#)(0) = 0 #
=Be car eful an d NOT mix up th e mu ltiplicat ion a nd a ddition r ules.
For example:
(5) + (2) = 7, bu t (5)(2) = 10
0 + 5 = 5, but 0 5 = 0
A nu mber x being mu ltiplied by 1 can be r epresen ted as follows
(excluding oth er n ota tions):
(x)(1) = (1)(x) = 1(x) = (1)x = 1x = (x) = x
N o t e : Notice th e last two on right ha nd side. A negat ive sign t o the left
of a pa ren th esis (or an y grouping symbol) may be seen as mu ltiplicat ion by
1 oras finding th e opposite of a n um ber.
( The key is used to repr esent mu ltiplicat ion. Todifferen tiat e it from the lett er , th e calculat or displays th e mu ltiplicat ion
using an asterisk: . To demonst ra te
Wh a t t o d o: On t h e Ca lcu la t or S creen :
What is 13 27?M
The Texas In stru ments gra phing calculat ors u nder sta nd implied
mu ltiplica tion, so a ll of th e following ar e th e same a s 13 27:
13 * 27
13(27)
(13)27
(13)(27)
(13)*(27)
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=A raised dot is OK for mu ltiplicat ion on paper , but NOT on a
ca lcula tor. The key is to ent er a decima l point ONLY.
The sa me can be sa id for oth er grouping symbols: { } and [ ]. These ha ve
differen t mea nings on a calculat or.
So if you need to ent er the following in your calculat or :
3[ (4 + 8) 2 ]
Use an additiona l set of par ent heses in place of th e squar e brackets:
3( (4 + 8) 2 )
GGGGGGGGGGGGGGGGGG
EXPONENTSFJ ust as m ultiplicat ion is an abbreviat ed form of repea ted a ddition,
Exponen ts ar e used to abbreviat e repeat ed multiplicat ion.
Mat h S y m b o l:
Using the nu mber 3 as a n example, th is is what is meant by repeated
multiplication:
3 3 3 3 3 = = 243
5 factors of 3
is rea d a s: th ree to th e fifth power or s imply: th ree to th e fifth .
Using a litt le algebra now:
xxx x =
n factors ofx
is read a s: x to th e n th power or simply: x to the n .
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F The expr ession: 3 3 3 3 3 is sa id to be inExpa n ded Form while is in Expon en t ia l Form ,Exponen t ia l
Notat ion , or Expon en t N ota t ion .
Some ter min ology:
xn This is th e Poweror Exponent
This is th e Base
is rea d: x ra ised to th e firstpower or x to th e first .
N o t e : If a nu mber or var iable is ra ised to th e first power, we can write
it with out th e 1, th at is: .
is rea d: x squar ed, x ra ised to the secondpower or x to th e second .
=Dont sa y: x two or x t o th e two.
is rea d: x cubed, x ra ised to the thirdpower or x to the th ird.
=Dont sa y: x th ree or x to the th ree. Similar war nings for t he r estbelow.
is rea d: x ra ised to the fouth power or x to the four th .
is rea d: x ra ised to the fifth power or x to th e fifth .
Oh is read : x ra ised to the y th power or just x to th ey.
( Use th e power key: to input an exponen t. Ifyou just wan t to squa re a value, use th e key. There is an option to
input a cube, but I find stu dent s dont ever use it , so dont worr y about it.
N o t e : On some calculat ors, t he power or exponen t keys m ay look
like:xy,yx , or a b . Also, looking a t your calculat or , th ere a re oth er
keys/fun ctions th at ha ve powers in th em, like: , , (am ong oth ers),
but we will discuss th ese in other Lessons.
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An o t h e r N o t e : When I a m describing keystrokes to studen ts, I say
power when I refer to this key: . For exam ple, I would sa y is:
th ree power four . It s a persona l choice. You can st ill just sa y: th ree to
th e four th power.
Lets do some calculator drills:
Wh a t t o d o: On t h e Ca lcu la t or S creen :
What is ?M
Or :
M
Of cour se, you could also just type:
M
What is negative th ree ra ised to
th e four th power?
=Since we ar e ra ising a negativenu mber t o a power, it MUSTbe
enclosed within par enth eses,
oth erwise, you ma y get th e wrong
an swer (depending on t he power).
So we want : ,NOT:
M
This is th e incorr ect way:M
is really th e opposite of .We get th e wrong an swer.
GGGGGGGGGGGGGGGGGG
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DIVISIONFDivision is th e inverse or r everse opera tion of multiplicat ion. It tells ushow man y times one n umber is cont ained with in an oth er n um ber.
Mat h S y m b o l:
The n um ber being divided is called th e Dividend. The num ber doing the
dividing is called th e Divisor. The r esu lt of a d ivision is called th e
Quot ien t .
Dividend Divisor = Quotient
To divide a nu mber x by a nu mber y, we can use a ny of the following
notations:
N ot a t ion : Com m en ts:
x yNot used very much in algebra . OK to use. Appear s in
opera tions dea ling with fractions. Grea t for n um bers.
x /y
or
OK to use, but ma y be a litt le vague wh en dea ling with
complicat ed expressions . Used typically to writ e fractions
in a m ore compa ct form , such a s in an exponen t or ma tr ix.
This is th e bestway to express division in a lgebra,
especially when t he item s get more complicat ed. The top
par t is called th e Numer a t o r an d the bott om par t is
called th e Denomina tor . Try to use th is nota tion wh en
possible.
or
y x
These a re used when perform ing long division and ar e
the same as:
If a n um ber does not exactly divide into another nu mber , th en t her e is a
R ema i nder .
F Dividend = (Divisor) (Quotient) + Remainder
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E XAMP LE 10: Usin g some nu mber s: 20 3 or is:
RULES OF DIVISION
R u l e 1 : positive positive = positive
nega t ive nega t ive = posit ive
The qu otient of num bers with like signs is positive.
R u l e 2 : posit ive nega t ive = nega t ive ornega t ive posit ive = nega t ive
The quotient of nu mber s with differen t s igns is n egative.
R u l e 3 : 0 an y non-zero nu mber = 0
R u l e 4 : an y non-zero num ber 0 = undefined.
FWe wont cons ider th e case. This is discus sed in
calculus.
These Rules can be abbreviat ed as:
=Dividing by zero is NOTa llowed! Zero cann ot be in t he
denomina tor of a fraction. Dont go there! Do not pa ss Go. Do not
collect $200. Oops! I got a litt le ca r ried a way t her e. Sor ry. L
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FRECIPROCALSF
Y The Reciproca l of a n um ber x is .
Y The reciproca l of a fraction is
N O
Y A nu mber mu ltiplied by its r eciprocal is always equa l to 1. Her e ar e a
couple of exam ples:
and
This is called t he Mul t ip l icat ive Inverse property.
Y The r eciproca l itselfis also called th e mu ltiplicat ive inverse.
Y Zero is the only nu mber t ha t does not h ave a reciprocal.
Y With r eciprocals, all divisions can now be re-writ ten as m ult iplica tions:
a b is th e sam e as: , or
( The key is used to per form divisions . It is
displayed on t he calculat or screen as a forwa rd slant : /. The r ecipr ocal ofa num ber can be foun d by using th e reciprocal key: . In pra ctice, th is
key is very ra rely used. Most stu dents just use: th en th e nu mber.
L One of th e places ma ny student s ma ke mistakes u sing
th e ca lcula tor is in t he a rea of divisions . Be extra careful
an y time you need t o perform th is opera tion or doing
fra ctions on th e calculat or.
HOT TIP!
When in doubt , just enclose th e entire num erator within an
extra set of par ent heses a nd do likewise with t he entire
denominator.
E XAMP LE 11:
To evalua te us ing th e ca lcula tor, visua lize th is as : .
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The express ion: sh ould be seen as:
( The last two expressions would be ent ered a s:
Wh a t t o d o: On t h e Ca lcu la t or S creen :
What is ?
M
What is ?
M
FNEGATIVES WITH DIVISIONF
Y The following ar e all equa l:
= = =
Y This equation is always t ru e:
FOTHER EQUATIONS WITH DIVISIONF
E qu a t ion : Com m en ts:
Any n um ber, over itself, is one.
A num ber, divided by 1, is just itself.
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ORDER+ OF OPERATIONSNow tha t we ha ve all these ar ithmetic operat ions, what ha ppens when we
combin e them? How do we evalu at e somet hin g like: 2 + 3 4 ? Do we add
the 2 and t he 3, then mu ltiply by 4, or do we mu ltiply th e 3 an d t he 4 firs t,then add 2 t o th is? Before we can an swer t his, we need to ta lk about th e
order of operat ions . We will st ar t t his discussion with group ing symbols.
FGr oup i ng Symbo l s ar e used t o show certa in ma th emat ical operat ionsshould be done before oth ers in an expression. Her e is a list of th e most
comm on symbols used in gr oup ing:
S ym bols: Com m en ts:
These a re th e most comm only used grouping symbols.E x a m p l e : 1 (2 + 3)
( Only use par ent heses for grouping.
or Squar e
Brackets
Used to enclose items th at already ha ve parent heses.
E x a m p l e : 4[1 (2 + 3)]
( Used for ma tr ices.
or Cu rlyBrackets
Used to group items th at ha ve squar e brackets.
E x a m p l e : 6 + {5 4[1 (2 + 3)]}
( Used t o enclose item s in lists.
or Vinculum
Used to group items th at ha ve braces.
E x a m p l e :
First, evalua te everything in t he nu mera tor.Next, evaluate
everything in t he denominator. Finally, divide th e
nu mera tor by the denominat or.
A detailed exam ple of th is will be shown lat er .
AbsoluteValue
Bars
Do everyth ing within t he bar s first , then ta ke its absolutevalue.
A detailed exam ple of th is will be shown lat er .
Do everyth ing inside th e ra dical fir st , then ta ke th e root.. Exa mples of th is ar e in th e Algebra Lesson: Rad icals.
F th e ra dical sign is rea lly just
Th e vin cu lu m is u sed t o ext en d t he sign :
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Ther e ar e oth er group ing symbols/operat ions , but we wont en coun ter
th em u nt il oth er lessons.
FALTERNATE GROUPING NOTATIONF
Say we ar e given somet hing t ha t looks like th is:
Which is ra th er complicat ed looking. Inst ead of using braces an d bar s, th e
par enth eses and brackets can altern at e, so th e previous problem would
look like:
OK, its st ill a mess, but it is a litt le eas ier on th e eyes.
L We dont have to use par ent heses. It s per fectly
acceptable t o write an y of th e following for grouping:
= = =
But its usu ally best to stick with pa ren th eses.
FORDER OF OPERATION RULESF
Y Eva lua te expressions with in groupin g symbols. So, if you h ave, say,
1 + (8 3), you would d o 8 3 first . If you ha ve items grouped with in
anotherset of grouping symbols, evalu at e first th e inn er set of grouping
symbols.
An exam ple of th is would be: 5 + 4[3 (1 + 2)]. You would eva lua te (1 + 2)
first , since it is th e inner most set of grouping symbols, then ta ke car e of
everyth ing inside of the br ackets.
Y Per form all exponen tial expressions before oth er ar ithm etic opera tions.
Given: , you need to squa re th e 4 first, NOT add t he 3 with th e 4.
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Y Next, per form a ll mu ltiplica tions and /or divisions
, but evaluat e th e expression in order from left-to-right .
To evaluat e 3 2 + 4 2, for examp le, multiply 3 2 firs t, since it is t o th e
left of 4 2.
This gives us: 6 + 4 2. Now, divide th e 4 by 2.
Our expression will now look like t his now: 6 + 2 .
Fina lly, do th e addition a nd t he r esult is 8 .
Y The last ru le is to do all additions an d/or subt ra ctions
but evaluat e th e expression in order from left-to-
right.
Lets eva lua te 8 3 + 5 1. Going from left-to-r ight, do 8 3 fir st . This
gives us 5:
5 + 5 1. Now add t he fives t o get 10:
10 1. Fin ally, do th e subt raction to get 9 as th e end result .
ORDER OF OPERATIONS
1Firs t, evalua te t he expression with in th e inn erm ost set of grouping
symbols.
2Next, evaluat e expressions t ha t ha ve exponen ts.
3Then , perform a ll multiplicat ions an d divisions , going from left-to-r igh t
in th e expression.
4Fin ally, do a ll additions a nd subt ractions, aga in, going from left-to-r ight
in th e expression.
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HOT TIP!
An a cronym t o help you rem ember th e above is:
P E M D A S
P: Pa rent heses
E: Exponen tsM: Multiplication
D: DivisionA: Addit ionS: Subtr action
PEMDAS can be m emorized eas ily if you remem ber t he followingmn emonic device:
Please Excuse My Dear Aunt Sally
FORDER OF PRECEDENCEF
Y The Or der of Precedence, is th e order of import an ce in per form ing
operat ions . The higher th e precedence, th e more import an t it is;
th erefore, n eeds t o be done before item s of a lower precedence. The order
is the sa me a s th e order of opera tions:
1 Groupin g symbols
2 Exponents
3 Multiplications / Divisions
4 Additions / Subtractions
There ar e additiona l opera tions with oth er orders of precedence, but th ese
will do for n ow.
N o t e : Most gra phing calculat ors an d compu ter progra ms follow th eOrder of Precedence ment ioned h ere. Sma ller, non-gra phing calculat ors
ma y evalua te expressions in a differen t way, so be car eful if usin g th em.
Consu lt your calculat ors Opera ting Man ua l for more inform at ion
Now th a t we ha ve all of th ese ru les, lets do severa l deta iled examples.
Fir st , by hand , showing all steps , then we will verify th e an swer by
evaluat ing th e expressions directly using th e calculat or.
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EXAMP LE 12: Simplify: 4 + 32 2 10
SOLUTION:Y There ar e no grouping symbols, so sta rt with t heexponent:
4 + 32 2 10
= 4 + 9 2 10
Y Next, do the mu ltiplicat ion:
4 + 9 x 2 10
= 4 + 18 10
Y Now, since we only ha ve additions a nd su btr actions, evalua te t he
express ion going from left-to-r ight:
4 + 18 10
= 22 10
Y Fina lly, do th e subt ra ction:
22 10
= 12
(Wh a t t o d o: On t h e Ca lcu la t or S creen :
Simplify: 4 + 32 2 10
M
E XAMP LE 13: Per form th e indicat ed opera tions:
5 + 2[3 4(7 6) + 22]
SOLUTION:Y Evaluat e wha t is in th e inn ermost parent heses first:
5 + 2[3 4(7 6) + 22]
= 5 + 2[3 4(1) + 22]
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Y Do th e Exponent next. The par enth eses around th e 1 ar e used for
implied mu ltiplicat ion, so th e exponen t is eva lua ted before th e
multiplication:
5 + 2[3 4(1) + 22]
= 5 + 2[3 4(1) + 4]
Y Per form t he mu ltiplicat ion within th e bra ckets :
5 + 2[3 4(1) + 4]
= 5 + 2[3 4 + 4]
Y Evalu at e th e subtr action a nd a ddition inside th e brackets (going from
left-to-right):
5 + 2[3 4 + 4]
= 5 + 2[1 + 4]
= 5 + 2[3]
Y The br ackets ar e now used a s implied mu ltiplicat ion, so perform th at
next:
= 5 + 2 [3 ]
= 5 + 6
Y Finally, add th e num bers together :
=5 + 6
= 1
This seems like we ar e doing a hu ge num ber of steps, but I am doing this
st ep-by-step, an d I re-display cert a in st eps for clar ity. Heres how th is
problem would look like if it were done by ha nd for a tes t ques tion:
5 + 2[3 4(7 6) + 22
]= 5 + 2[3 4(1) + 2
2]
= 5 + 2[3 4(1) + 4]
= 5 + 2[3 4 + 4]
= 5 + 2[1 + 4]
= 5 + 2[3]
= 5 + 6
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= 1If your inst ru ctor a llows you to combine st eps, th e above problem can be
done in even fewer st eps, but I only recomm end you do th is after a greatdea l of pr actice .
5 + 2[3 4(7 6) + 22]
= 5 + 2[3 4(1) + 4]
= 5 + 2[3 4 + 4]= 5 + 2[3]
= 5 + 6
= 1
(Wh a t t o d o: On t h e Ca lcu la t or S creen :
Per form th e indicat ed opera tions:
5 + 2[3 4(7 6) + 22]M
Remember t o use par enth eses only, not br ackets, when
ent ering th e keystrokes into your calculat or.
E XAMP LE 14: Simplify the expression usin g the order of operat ion
rules:
23
| 5 2 8|
SOLUTION:Y The a bsolute valu e bar s serve as grouping symbols, so
evaluat e what is inside them first :
23
|5 2 8|
Y Multiplicat ion ha s h igher precedence:
23
| 5 2 x 8 |
= 23
| 5 16 |
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Y Now do th e subt ra ction with in th e absolute value:
23
| 5 16 |
= 23
| 11 |
Y Next, perform t he absolute value: | 11| = 11
23
|11 |
= 23
11
Y Then, evaluat e th e exponent :
23
11
= 8 11
Y Fina lly, do th e subt ra ction:
8 11
=3
(Wh a t t o d o: On t h e Ca lcu la t or S creen :
Simplify: 23 | 5 2 8|M
y
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=
Y Evalu at e all subt ra ctions a nd a dditions in th e num era tor, going from
left-to-right:
= =
Y In t he denominat or, evaluat e the subtr action within th e parent heses
first:
=
Y Evaluat e the exponent next:
=
Y Now, do th e subt ra ction:
=
Y Fina lly, divide the n um era tor by th e denominat or (leaving it in
fractional form ):
=
You could h ave a lso just writt en 0.5 as th e final answer.
An a ltern at e way to evalua te t he original pr oblem is t o simplify the
numera tor a nd th e denominator a t t he same time. Star ting with:
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Y Exponent iate th e items in th e num erat or a nd do th e subtra ction in th e
denominator:
Y Do th e subtra ction in th e num erat or, and squa re th e 2 in t hedenominator:
Y Per form th e addition on top, an d th e subt ra ction below:
Y Fin a lly, divide th e top by the bott om (leaving it in fra ctiona l form):
=
( Be sur e to enclose th e entire numerator anddenominat or with in an extr a set of par enth eses.
In oth er words , cha nge: into:
Wh a t t o d o: On t h e Ca lcu la t or S creen :
Simplify:
M
Convert to a fra ction:
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LESSON 3 QUIZWhen doing th ese problems, tr y to also do th em u sing your calculat or , (if
possible) to get more pr actice using it.
1Write t he following expressions without absolut e value bar s,simplifying a lso, if possible:
Y
Y
Y
Y
Y
2 Rewrite t he following without absolut e values, leaving t he
an swer in E XACT form:
Y
Y
Y
3Fin d t he opposites of the following:
Y 5. The opposit e is: _____
Y 8. Th e opposit e is: _____
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Y . The opposit e is: _____
Y . The opposit e is: _____
Y x. The opposit e is : _____
4Per form t he following opera tions:
Y 8 + (2) =
Y 8 + 2 =
Y 8 + (2) =
Y Subt ra ct 2 from 8:
Y 8 2 =
Y 8 2 =
Y 8(2) =
Y (8)( 2) =
Y 8 2 =
Y =
Y =
Y Find th e squa re of negat ive five =
Y =
Y =
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5What is the distan ce between 5 and 7?
6What is the dista nce between x and y?
7Which of th e following is cons idered as th eBESTway to input two times t hr ee in a calculat or?
(2)(3) 2(3) (2)3 2 3
(2) (3) 2 (3) (2) 3
8What is th e expanded form of ?
9Write th e Exponen tial Notat ion for 7 7 7:
blWhat is th e base of ?
bmWhat is th e exponen t of ?
bnWhich operation is performed firstfor: ?
boWhich of the following a re not typical grouping symbols used in ma thexpressions?
bpWhat is PEMDAS?
bqTru e or F alse. Subtr action h as a higher order of precedence th anDivision. ________
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brSimplify. Show all steps. Write work on a separ at e sheet of paper.
Y 3 2 + 8 =
Y (2 + 3) 5 =
Y (6 2)(8 + 1) =
Y
Y
Y
Y
Y
Y
=
Y
Y
ANSWERSONNEXTPAGE
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ANSWERS
1Write t he following expressions without absolut e value bar s,simplifying a lso, if possible:
Y 8
Y 7.2
Y 6
Y 10 16 = 6
Y 5
2 Rewrite t he following with out absolut e values, but leaving t he
an swer in E XACT form:
Y f 2
E x p l a n a t i o n : since 2 f is negat ive, you wa nt to get a positive value,since absolut e values a re a lways positive, so th e way a round th is is to just
switch th e places of th e nu mber s.
Y f A7
E x p l a n a t i o n : since is alr ead y positive (t ry it on a calculat or ), you
just need to remove th e absolute value bar s.
Y 3 A7
E x p l a n a t i o n : Similar to first Y in Problem 2. These t ypes of quest ionsare very typical on t ests.
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3Fin d t he opposites of the following:
Y 5. The opposite is: 5
Y 8. The opposite is: 8
Y . The opposite is: A3
Y . The opposite is: 6
Y x. The opposite is:x
4Per form t he following opera tions:
Y 8 + (2) = 6
Y 8 + 2 = 6
Y 8 + (2) = 10
Y Subt ra ct 2 from 8:
This is wr itt en as : 8 (2) = 8 + 2 = 6
Y 8 2 = 8 + (2) = 10
Y 8 2 = 16
Y 8(2) = 16
Y (8)( 2) = 16
Y 8 2 = 4
Y = 4
Y = 4 4 4 = 64 (or just use calculat or to get th is value).
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Y Find th e squa re of negat ive five = 25
N o t e : The an swer isNOT25, since we wan t: .
A typical err or is to writ e wha t wa s a sked for as:
Y = 1
Y = 301
5What is the distan ce between 5 and 7?
Using th e distan ce form ula, we get: 12
F We could have also writt en: 126What is the dista nce between x and y?
Using th e distan ce form ula a gain, we get: y x
F We could have also writt en: x y7Which of th e following is cons idered as t he BESTway to inpu t two
times t hr ee in a calculat or?
(2)(3) 2(3) (2)3 D 2 3(2) (3) 2 (3) (2) 3
F All of th e above ar e acceptable. I wan t you t o useth e one t ha t would involve th e fewestkeyst rokes; however, if you needto add more keystr okes so th at th e expression seem s clearer to you,
th en go ah ead an d add more.
8What is th e expanded form of ?A n s w e r :6 6 6 6
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9Write th e Exponen tial Notat ion for 7 7 7:
A n s w e r :73
blWhat is th e base of ?A n s w e r :The base is 5. The base isNOT5. By order of pr ecedence,
the squar e is justwith th e 5, an d does not include th e negative sign. If
we want ed th e base to be 5, th en t he expression sh ould ha ve been
written: .
bmWhat is th e exponen t of ?A n s w e r :The exponen t is 5.
bnWhich operation is performed firstfor: ?
A n s w e r :The Exponen t or Squar e the Four. Exponen ts ar e done beforeth e oth er operat ions.
boWhich of th e following ar e not t ypical group ing symbols used in m athexpressions?
D
bpWhat is PEMDAS?A n s w e r : It is a n a cronym t o help you r emember th e order of opera tions
of rea l nu mbers . The letter s sta nd for:
P: Pa rent hesesE: Exponen tsM: Multiplication
D: DivisionA: Addit ion
S: Subtraction
bqTru e or F alse. Subtr action h as a higher order of precedence th anDivision.
A n s w e r :FALSE. Subt ra ction (along with addition) ha s t he lowest
order of precedence present ed in t his lesson.
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brSimplify. Steps ar e sh own below each pr oblem.
Y 3 2 + 8 = 2
A n s w e r : 3 x 2 + 8
=6 + 8
= 2
Y (2 + 3) 5 = 25
A n s w e r : (2 + 3) 5
= (5) 5
= 25
Y (6 2)(8 + 1) = 36
A n s w e r : (6 2)(8 + 1)= (4)(9)
= 36
Y 4
A n s w e r : 2(1 3 + 2 2)
= 2(1 3 + 22)
= 2(1 3 + 4 )
= 2(1 3 + 4)
= 2(2 + 4)
= 2(2 + 4)
= 2(2 )
= 4
Y 12
A n s w e r : 10 [2 + (4 23)]
= 10 [2 + (4 23)]
= 10 [2 + (4 8 )]
= 10 [2 + (4 8)]
= 10 [2 + (4)]
= 10 [2 + (4)]
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= 10 [2]
= 10 + 2
= 12
Y 64
A n s w e r :
=
= 64
Y 6
A n s w e r : 12 + 2[4 + (2 3)2
]= 12 + 2[4 + (1)2]
= 12 + 2[4 + (1)2]
= 12 + 2[4 + 1 ]
= 12 + 2[4 + 1]
= 12 + 2[3]
= 12 + 2[3]
= 12 + (6)
= 6
Y 97
A n s w e r : | 3| + (8 + 2)2
= (3) + (10 )2
= (3) + (10)2
= (3) + 100
=3 + 100
= 97
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Y = 20
A n s w e r :
=
=
=
=
=
=
=
=
=
=
=20
Y 4
A n s w e r :
=
=
=
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=
= 4
Y 5
A n s w e r :
=
=
=
=
=
=
= 11 (6)
= 11 + 6
=5